Observational quantification of three-dimensional anisotropies and scalings of space plasma turbulence at kinetic scales
Tieyan Wang, Jiansen He, Olga Alexandrova, Malcolm Dunlop, Denise Perrone
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Observational quantification of three-dimensional anisotropies and scalings of space plasmaturbulence at kinetic scales
Tieyan Wang , Jiansen He, Olga Alexandrova, Malcolm Dunlop,
4, 1 and Denise Perrone RAL Space, Rutherford Appleton Laboratory, Harwell Oxford, Didcot OX11 0QX, UK School of Earth and Space Sciences, Peking University, Beijing 100871, China LESIA, Observatoire de Paris, Universit´e PSL, CNRS, Sorbonne Universit´e, Univ. Paris Diderot, Sorbonne Paris Cit´e, 5 place JulesJanssen, 92195 Meudon, France School of Space and Environment, Beihang University, Beijing 100191, China ASI - Italian Space Agency, via del Politecnico snc, 00133 Rome, Italy (Accepted June 5, 2020)
Submitted to ApJABSTRACTA statistical survey of spectral anisotropy of space plasma turbulence is performed using five yearsmeasurements from MMS in the magnetosheath. By measuring the five-point second-order structurefunctions of the magnetic field, we have for the first time quantified the three-dimensional anisotropiesand scalings at sub-ion-scales ( <
100 km). In the local reference frame ( ˆ L ⊥ , ˆ l ⊥ , ˆ l (cid:107) ) defined withrespect to local mean magnetic field B (Chen et al. 2012), the statistical eddies are found to bemostly elongated along B and shortened in the direction perpendicular to both B and local fieldfluctuations. From several d i (ion inertial length) toward ∼ d i , the ratio between eddies paralleland perpendicular lengths features a trend of rise then fall, whereas the anisotropy in the perpendicularplane appears scale-invariant. Specifically, the anisotropy relations for the total magnetic field at 0.1-1.0 d i are obtained as l (cid:107) (cid:39) . · l . ⊥ , and L ⊥ (cid:39) . · l . ⊥ , respectively. Our results provide newobservational evidence to compare with phenomenological models and numerical simulations, whichmay help to better understand the nature of kinetic scale turbulence. Keywords: turbulence, magnetic field INTRODUCTIONThe energy distribution at a certain scale (or k space) is known to be not isotropic in the turbulence of magnetizedplasma, also known as spectral anisotropy (Cho & Vishniac 2000). This particular feature reflect the preferentialdirection of the energy cascade with respect to the local background magnetic field B (Podesta 2009). Most of ourexperimental knowledge of space plasma anisotropy comes from in-situ observations made within the solar wind (SW),which is a nearly collisionless plasmas stream released from the Sun (Bruno & Carbone 2013).At large magnetohydrodynamic (MHD) scales, the pattern of correlation function for the magnetic field at 1 AUhas two major components referred to as “Maltese cross”, exhibiting elongations in both parallel and perpendiculardirection with regard to B (Matthaeus et al. 1990). This signature is summarized as the “slab+2D” model, whichassumes no specific nature of the fluctuations but just describe the fluctuations as a combination of waves with k (cid:107) andstructures with k ⊥ . Another type of anisotropy model is based on “critical-balance (CB)” conjecture, where the keyhypothesis relies on the comparable scale of the linear Alfv´en time and turbulence non-linear time in a vanishing cross-helicity system (Goldreich & Sridhar 1995). As a result, the spectral anisotropy scales as k (cid:107) ∝ k / ⊥ . By introducingthe idea of “dynamic alignment” between magnetic and velocity field fluctuations, Boldyrev (2006) modified the non-linear time and established the 3D anisotropic turbulence model, where the eddies have three different coherent scales. Corresponding author: Tieyan [email protected] a r X i v : . [ phy s i c s . s p ace - ph ] J un Wang et al.
Indeed, numerous observations have found agreement between measurements and CB theories (Horbury et al. 2008;Luo & Wu 2010; Chen et al. 2011, 2012). Despite this consistency, recent revisit of anisotropy in the solar wind havereported some puzzling results and raised more concerns to be considered, such as intermittency (Wang et al. 2014;Pei et al. 2016; Yang et al. 2017), the discrepancy between velocity field and magnetic field anisotropy (Wicks et al.2011; Wu et al. 2019a,b; Yan et al. 2016), dependence on the heliocentric distance (He et al. 2013) and solar windexpansion (Verdini et al. 2018, 2019). Moreover, the 3D self-correlation functions are shown to be isotropic in (Wanget al. 2019b; Wu et al. 2019a,b). Therefore, the anisotropic nature of the solar wind at MHD scales is still an openquestion.At kinetic scales, the turbulence still remains or become much anisotropic (i.e., Chen et al. (2010); Oughton et al.(2015) and references therein). The standard Kinetic Alfv´en Wave (KAW) turbulence model, also on basis of CBconjecture by assuming the linear KAW propagation time to be comparable to nonlinear time, predicts an anisotropyscaling of k (cid:107) ∝ k / ⊥ (Howes et al. 2008; Schekochihin et al. 2009). It is also important to pay attention to the muchcomplicated physical process at kinetic scales than at MHD regime due to plasma kinetic effects (see the review byAlexandrova et al. (2013) and Alexandrova et al. (2020)). For example, the modified KAW turbulence model withintermittent 2D structures has k (cid:107) ∝ k / ⊥ (Boldyrev & Perez 2012). Zhao et al. (2016) suggested a model for kinetic-scaleAlfv´enic turbulence which incorporate the dispersion and intermittency effects. Boldyrev & Loureiro (2019) consideredthe decisive role played by the tearing instability in setting the aspect ratio of eddies and hence predicted the spectralanisotropy scalings between k (cid:107) (cid:46) k / ⊥ and k (cid:107) (cid:46) k ⊥ . Most recently, Landi et al. (2019) proposed a phenomenologicalmodel considering the intermittent two-dimensional structures in the plane perpendicular to B . In their model, theprescribed perpendicular aspect ratio of these structures could determine the anisotropy as k (cid:107) ∝ k / α +1) ⊥ , where α isproportional to the space-filling of the turbulence.In recent high resolution three-dimensional kinetic simulations, the spectral anisotropy has received considerableattentions (Groˇselj et al. 2018; Franci et al. 2018; Arzamasskiy et al. 2019; Cerri et al. 2019; Landi et al. 2019). Basedon different methods of measuring the anisotropy, dissimilar scaling relations have been found (i.e., k (cid:107) ∝ k / ⊥ inGroˇselj et al. (2018) and k (cid:107) ∝ k ⊥ in Arzamasskiy et al. (2019). Specifically, for the analysis based on multi-pointlocal structure functions ( SF ) which will be introduced in Section 2, the anisotropy tends to become “frozen” whenapproaching ion scales (i.e., k (cid:107) ∝ k . ⊥ in Landi et al. (2019)). By comparing the results from three different simulationsincluding the hybrid particle-in-cell (PIC), Eulerian hybrid-Vlasov, and fully kinetic PIC codes, Cerri et al. (2019)found that the anisotropy scalings tend to converge to l (cid:107) ∝ l / ⊥ based on a unified analysis of five-point SF s.The Earth’s magnetosheath (MSH) offer a unique lab different from SW, such as enhanced compressibility, intermit-tency, as well as the kinetic instabilities (Alexandrova 2008). In addition, the spacecraft measurements in MSH tendto cover a wider range of angle between bulk flow velocity and B as compared with solar wind (He et al. 2011), thusallowing us to diagnose 3D nature of the fluctuations in a relatively short interval. Using Cluster measurements, Man-geney et al. (2006) found the strong anisotropy of the electromagnetic fluctuations, with k ⊥ extending for two decadeswithin the kinetic range kd e ∈ [0 . , kρ i = 3 . k ⊥ (cid:29) k (cid:107) ) above the spectral break in the vicinity of ion scale. In addition, due to the Doppler shift, the magneticfluctuations have more energy along the B × V direction in their analysis. Similar 2D turbulence at kinetic scales wasobserved in a recent statistical study of magnetic field turbulence in the solar wind (Lacombe et al. 2017). He et al.(2011) computed the spatial correlation functions of both magnetic field and density fluctuations in the 2D ( l (cid:107) , l ⊥ )plane, it is shown that the turbulence close to ion scales is comprised of two populations, where the major component ismostly transverse and the minor one is oblique. Using measurements from Magnetospheric Multiscale mission (MMS)(Burch et al. 2016), Chen & Boldyrev (2017) studied the two-point SF of the magnetic field in the same plane andprovided evidence of strong anisotropy at smaller scales (11 < kρ i < kρ i ∼ SF to statistically quantify the 3D anisotropy of the magnetic turbulence at sub-proton scale ( <
100 km). The new observational evidence we obtained, such as the empirical relations of the anisotropy scalings, can nisotropies and scalings DATA AND METHODSThe burst mode data from four MMS spacecraft, including magnetic field (128 Hz) from FIELD instrument (Russellet al. 2016) and ion moments (6.7 Hz) from FPI instrument (Pollock et al. 2016) are used in this study. 349 MSHintervals from September 2015 to December 2019 have been selected for the statistical analysis, whereas a 10 minutesevent on 4 October 2017 is presented to show a typical event with anisotropy signatures.To quantify the anisotropy, we use the five-point second-order SF in this study. Compared with two-point SF ,five-point SF is more suitable for studying spectral anisotropy at sub-ion regime (Cho 2019; Landi et al. 2019). Moredetails of the difference between multi-point SF s are provided on APPENDIX A.The five-point SF is the ensemble average of the squared variation ∆ f from 5-point, as function of displacement l ,the S (5)2 ( l ; f ) is defined as S (5)2 ( l ; f ) = (cid:104)| ∆ f ( r , l ) | (cid:105) r (1)The spatial variation from 5-point is measured as∆ f ( r , l ) = [ f ( r − l ) − f ( r − l ) + 6 f ( r ) − f ( r + l )) + f ( r + 2 l )] / √
35 (2)Where ∆ f can represent perpendicular, parallel, or total magnetic field fluctuations, (cid:104) ... (cid:105) r is ensemble average overpositions r .By studying the three-dimensional distribution of the S (5)2 ( l ; f ) with respect to l , the statistical shape of the eddiesin turbulence can be thus inferred from the contours of S (5)2 ( l ; f ).In the computation, the velocity field is used to link the time scale with space displacement as l = τ · V according toTaylor hypothesis (Taylor 1938), which has been tested in In APPENDIX B to be valid for most of our events. Here V can be adopted as the mean velocity V mean during the interval of interest or considered as the local velocity field V local , where V is interpolated on the resolution of B in the calculation. Most of previous works performed in solarwind and magnetosheath use V mean for simplicity (i.e., Chen et al. (2010, 2012); Chen & Boldyrev (2017); Wang et al.(2019b); Wu et al. (2019a)). Also, the study of electron scale magnetic field structure functions is based on V mean (Chen & Boldyrev 2017). In a recent paper by Verdini et al. (2018) the authors have considered the effects of localvelocity by using V local in their analysis of velocity filed structure functions. We use five-point V local in this paper,which is defined as V local = [ V ( r − l ) + 4 V ( r − l ) + 6 V ( r ) + 4 V ( r + l )) + V ( r + 2 l )] /
16. For the small spatial scalesconsidered here, let us rewrite l = τ · V as l = τ · ( V + δ V ). Since τ is small, the major contribution of velocity termis from the large scale mean velocity V . In other words, the resolution of V is not the key factor for interring l , sinceit only determines δ V , whose amplitude is generally much smaller than V in the magnetosheath environment underthis study. To verify this point, we have performed and compared the analyses on basis of V mean and V local , and theresults turn out to be nearly identical. Hence, we justify that the effects of interpolating V at the time points of B ,or only considering V mean is negligible in the analysis of small-scale magnetic field structure functions.Once the SF with respect to l is computed, it can be projected into local coordinates with respect to the localmagnetic field B local as in Chen et al. (2012); Verdini et al. (2018) to study its 3D features. This coorinates allowsus to compare the results with recent simulations on basis of the same reference frame (Landi et al. 2019; Cerri et al.2019), and is consistent with previous choice of studying spectral anisotropy at various scales (Chen et al. 2012; Chen& Boldyrev 2017; Verdini et al. 2018; Wu et al. 2019a). In the Cartesian coordinates system ( ˆ L ⊥ , ˆ l ⊥ , ˆ l (cid:107) ), ˆ l (cid:107) is along B local = [ B ( r − l ) + 4 B ( r − l ) + 6 B ( r ) + 4 B ( r + l )) + B ( r + 2 l )] /
16, ˆ L ⊥ is the ”displacement direction” along δ B ⊥ = B local × [ δ B × B local ], and ˆ l ⊥ = ˆ l (cid:107) × ˆ L ⊥ . The Cartesian system can be also converted into spherical polarcoordinates system as ( l, θ B , φ δB ⊥ ), where θ B represent the angle between B local and l , φ δB ⊥ represent the anglebetween L ⊥ and the projection of l on the plane perpendicular to B local . Similar to Chen et al. (2012), angles greaterthan 90 ◦ are reflected below 90 ◦ to improve scaling measurements accuracy. Specifically, by setting the ranges of θ B Wang et al. and φ δB ⊥ , the SF in the three orthogonal directions can be obtained as S (5)2 ( L ⊥ ; f ) ≡ S (5)2 ( L ⊥ ; f, ◦ < θ B < ◦ , ◦ < φ δB ⊥ < ◦ ) (3) S (5)2 ( l ⊥ ; f ) ≡ S (5)2 ( l ⊥ ; f, ◦ < θ B < ◦ , ◦ < φ δB ⊥ < ◦ ) (4) S (5)2 ( l (cid:107) ; f ) ≡ S (5)2 ( l (cid:107) ; f, ◦ < θ B < ◦ , ◦ < φ δB ⊥ < ◦ ) (5)By equating the value between pairs of S ( l ⊥ ), S ( l (cid:107) ), and S ( L ⊥ ), we could infer the anisotropy relation between l ⊥ , l (cid:107) , and L ⊥ . EXAMPLE OF LOCAL 3D TURBULENCEHere we present an example with clear signatures of 3D anisotropy. The event is observed downstream of the quasi-parallel shock during 08:02:13-08:12:33 on 4 October 2017. Figure 1 shows the overview of the event. As plottedin Figure 1a-b, the magnetic field is around 6.08 (0.61, 0.06, 0.79) ± (4.1, 3.6, 3.8) nT, exhibiting numerous largedirectional changes while keeping its magnitude. In contrast, the ion velocity is quite stable at 288 (-0.88, 0.46, 0.07) ± (15.5, 10.2, 13.5) km/s. Figure 1c shows the instantaneous increment of the total magnetic energy δB as a functionof scale and time. The magnitude of δB generally increase with the increase of scales and is changing intermittentlywith time. Similar to δB , the instantaneous increments of the perpendicular energy δB ⊥ and parallel energy δB (cid:107) alsoexhibit the same trend with respect to spatial scale as seen in Figure 1d-e, while the former one is stronger. Figure1f-g plots the corresponding θ B and φ δB ⊥ , respectively. Due to the rapid rotations of magnetic field, the distributionof θ B and φ δB ⊥ covers a wide range within (0 , π ) during the whole interval, thus allowing us to infer the 3D anisotropyof magnetic turbulence with sufficient data points. Figure 1.
Event overview. a) Three components and the strength of the magnetic field. b) Three components of velocity field.c-e) Instantaneous total, perpendicular, and parallel magnetic energy as a function of scale and time. f) Instantaneous anglebetween local magnetic field and space displacement vector l , as a function of scale and time. g) Instantaneous angle between L ⊥ and the projection of l on the plane perpendicular to local magnetic field. nisotropies and scalings SF s are obtained from four MMS spacecraft, then binned and averaged. Each bin is required to have a minimumnumber of 200 data points to ensure reliable results as in Chen et al. (2010). For the SF of the total magnetic fieldenergy as projected in the ( l (cid:107) , (cid:112) l ⊥ + L ⊥ ) plane, the contours of S (5)2 ( l ; B ) are elongated in the parallel directions(Figure 2a), where the values at perpendicular direction are much larger than the ones in the parallel direction (i.e. S (5)2 ( l ; B ) at l ⊥ = 60 km is more than 100 times larger than the one at l (cid:107) = 60 km). This signature indicate sub-ion-scale ( l < d i ) anisotropy with k ⊥ (cid:29) k (cid:107) , and is in agreement with Chen & Boldyrev (2017). Furthermore, we findthat the contours of S (5)2 ( l ; B ) are also elongated in the “displacement” direction as seen in the ( l ⊥ , L ⊥ ) plane (Figure2b), suggesting the three-dimensional characteristics of the anisotropy. In addition to S (5)2 ( l ; B ), we also consider thecontribution of SF by the perpendicular and parallel magnetic field, namely the S (5)2 ( l ; B ⊥ ) and S (5)2 ( l ; B (cid:107) ). As seenin Figure 2c-d, the pattern for the contours of S (5)2 ( l ; B ⊥ ) is nearly comparable to the ones of S (5)2 ( l ; B ) in Figure2a-b. But the contours in Figure 2e are flatter than the ones in Figure 2c, meaning the anisotropy of S (5)2 ( l ; B (cid:107) ) isslightly stronger than S (5)2 ( l ; B ⊥ ) in the ( l (cid:107) , (cid:112) l ⊥ + L ⊥ ) plane. The much-elongated compressive fluctuations along B is consistent with solar wind observations in Chen et al. (2010, 2011, 2012), which may imply the less damped stateof the more anisotropic fluctuations. On the contrary, the contours of S (5)2 ( l ; B (cid:107) ) are roughly isotropic in the ( l ⊥ , L ⊥ )plane, meaning the absence of gradient in the perpendicular plane for the parallels fluctuations.Let us draw the attention of the reader to the point that, the sampling of SF s in the perpendicular and paralleldirection is dissimilar as seen in Figure 1f, whereas the parallel data are much discretely distributed. To test whetherthe stationarity of the sampling will have an effect on the results of
SF s , we have divided the time-series into two subintervals and analyse the
SF s separately. During Interval 1 (08: 02: 13 - 08: 07: 13 UT), the measurements alongparallel directions (i.e., θ B < ◦ ) are less than the ones at oblique directions ( θ B > ◦ ). In contrast, during Interval 2(08: 07: 13 - 08: 12: 33 UT), the measurements along parallel directions are much frequent and the overall samplingare more homogeneous than Interval 1. As expected, the SF s for Interval 1 have no measurements within the range of60 km < l (cid:107) <
100 km, 0 km < (cid:112) l ⊥ + L ⊥ <
18 km), while the SFs for Interval 2 cover the complete wavenumber space.More importantly, the extension feature of the contours along l (cid:107) direction in these two sub-intervals appears quitesimilar as compared with the results from the whole interval. Hence, the anisotropic features of the turbulence couldbe viewed as stationary regardless of the interval selection. In fact, as the first step of computing SF s , we calculate thetime difference between continuous sampling points rather than discontinuous sampling points. As the second step,the calculated time differences satisfying certain θ B conditions are collected together from discretely(discontinuously)distributed time points. This discontinuous collection will not significantly influence the analysis results as long as thewhole time interval is statistically time stationary.To inspect the scale-dependency of the anisotropy more precisely, we have computed the SF in the three orthogonaldirections as defined by equation (3)-(5). For the 1D SF of the total magnetic field shown in Figure 2g, the relationof S ( l ⊥ ) > S ( L ⊥ ) > S ( l (cid:107) ) are satisfied at all scales as expected, thus confirming the 3D nature of anisotropy again.Moreover, this anisotropy is found to be scale-dependent. For example, at energy level of 0.01 nT , the perpendicularlength of the “statistical eddy, l ⊥ ∼ L ⊥ ∼ l (cid:107) ∼
35 km. As the energy level increase to 1 nT , l ⊥ , L ⊥ , and l (cid:107) becomes approximately 60km, 90 km, 150 km, respectively. The change of l ⊥ : L ⊥ : l (cid:107) ratio from 0.17: 0.23: 1 to 0.4: 0.6: 1 suggests that asscales increase, the anisotropy between parallel and perpendicular lengths becomes weak, while the anisotropy betweentwo perpendicular lengths in the perpendicular plane remains almost unchanged. Setting an energy range as [0.01, 0.5]nT , the SF s could be fitted by the power laws as l . ⊥ , L . ⊥ , and l . (cid:107) , where the standard error of the mean is 0.09,0.08, 0.10, respectively. The power law index of the second-order structure function, g , is usually related to the powerspectral index, α , by α = g + 1 (Chen et al. 2010). Hence the spectral indices in the three directions are 2.92, 2.82,and 4.08, respectively. The perpendicular spectral indicies are close to 8/3 but steeper than 7/3, which are consistentwith previous findings both in the MSH and SW (Alexandrova et al. 2008; Huang et al. 2014; Matteini et al. 2017;Chen & Boldyrev 2017; Alexandrova et al. 2009). For the SF s of δB ⊥ , the trends plotted in Figure 2i are essentiallythe same compared with the results of δB in Figure 2g, suggesting a dominant contribution of perpendicular magneticfield fluctuations to SF s. This point agrees with the “variance anisotropy found in Chen et al. (2010) and is againsupported by examining SF s of δB (cid:107) in Figure 2k, whose magnitudes are weaker than SF s of δB ⊥ in Figure 2g.Figure 2h displays the anisotropy relations for δB , where the blue curve represent l (cid:107) vs l ⊥ and the red curve represent L ⊥ vs l ⊥ . On one hand, as the perpendicular scales decrease from 1.0 d i to under 0.05 d i , the ratio of l (cid:107) /l ⊥ first Wang et al. increase and reached a maximum of ∼ ∼ d i , whereas the anisotropy scaling obeys l (cid:107) ∝ l . ⊥ . Then the ratioof l (cid:107) /l ⊥ begins to decrease and finally approaches 1 at ∼ d i . On the other hand, the ratio of L ⊥ /l ⊥ keeps steadyaround 1.5, obeying L ⊥ ∝ l . ⊥ . As expected, the anisotropy relations for δB ⊥ as shown in Figure 2j is quite similarwith the relations in Figure 2h, following l (cid:107) ∝ l . ⊥ and L ⊥ ∝ l . ⊥ . Nevertheless, the anisotropy relations for δB (cid:107) asshown in Figure 2l are dissimilar, following l (cid:107) ∝ l . ⊥ and L ⊥ ∝ l . ⊥ . Figure 2.
The SF s in the 2D plane, together with the 1D SF s and their anisotropy scalings. The left two columns include:(1) 2D SF s as a function of ( l (cid:107) , (cid:112) l ⊥ + L ⊥ ), and (2) SF s as a function of ( l ⊥ , L ⊥ ). The right two columns plot respectively: (1)1D SF as a function of l ⊥ , L ⊥ , and l (cid:107) , and the relations between l (cid:107) , l ⊥ , and L ⊥ . For each panel, the first, second, and thirdrows represent SF s of the total, perpendicular, and parallel magnetic field, respectively.4. STATISTICAL ANALYSIS OF THE ANISOTROPY SCALINGSIn this section, the sub-ion-scale anisotropy relations are investigated comprehensively based on a statistically surveyof 349 intervals during 2015-2019, when MMS instruments was in burst mode. These intervals are tagged as “mag-neotosheath on the MMS science data center website. In addition, to avoid the influence of shock or magnetopuase,the ion and electron energy spectrogram have been checked by eye to make sure they exhibit typical broadband MSHsignatures and are time stationary. As a result, 349 intervals with an average duration of ∼ β p , ion inertial length d i , and proton gyroradius ρ p ,cover the range of [60, 1680] s, [0.3, 80], [15, 130] km, and [45, 370] km respectively. The mean value of temperatureanisotropy T p ⊥ /T p (cid:107) is 1.05. The distribution of mirror mode threshold Σ mirror = T p ⊥ /T p (cid:107) − /β p ⊥ −
1, is also shownin Figure 3e. Since most of the values are negative, the influence of mirror instability is not strong in our database. nisotropies and scalings Figure 3.
Histograms of the events duration and plasma characteristic parameters.
Figure 4 presents the statistical results of the anisotropy relations. Concerning the parallel − perpendicular anisotropyof the total magnetic field δB , as revealed by the unique feature of the superimposed results in Figure 4a, a largeproportion of events exhibit analogous trend. For the median value of the data, when the scales decrease from 10 d i to0.01 d i , the anisotropy level as reflected from the vertical deviation from the isotropy reference line, displays a trend ofrise then fall, with the break point occurring near 0.1 d i . The fit of an empirical anisotropy relation l (cid:107) = a · l α ⊥ yield a of 2.44 and a scaling of α l = 0 . ± .
03 at large scales within [0.1, 1] d i . Compared with three reference scaling-laws of α = 1 /
3, 2 /
3, and 3 /
3, the fitted scaling is closer to 2 /
3. At smaller scales within [0.04, 0.08] d i , we find a = 170, and α s = 2 . ± .
35. Likewise, the anisotropy relations of δB ⊥ and δB (cid:107) in Figure 4b-c display similar trends as Figure4a, where the anisotropy scalings at [0.1, 1.0] d i are obtained as α l = 0 . ± .
03 and α l = 0 . ± .
04 and the scalingsat [0.04, 0.08] d i are α s = 2 . ± .
32 and α s = 2 . ± .
14, respectively. By examining the distribution of power-lawscalings for each individual event as shown in the histograms in Figure 4, we confirm that for the scalings at [0.1, 1.0] d i (dark grey), a summary of over 100 events have a scaling centered near 2 / d i , the scalings are broadly distributed within [0, 4].Regarding the anisotropy of δB and δB ⊥ in the perpendicular plane, Figure 4d-e show that for most of the data,although L ⊥ > l ⊥ , the anisotropy level is stable since the scalings are close to 1. In addition, the empirical relation L ⊥ = b · l β ⊥ for the median value are fitted as L ⊥ = 1 . · l . ± . ⊥ , L ⊥ = 2 . · l . ± . ⊥ at [0.02, 1.0] d i , respectively.Lastly, the anisotropy for δB (cid:107) vanishes and follows a nearly isotropic relation of L ⊥ = 1 . · l . ± . ⊥ . CONCLUSION AND DISCUSSIONIn this paper, we have conducted a statistical survey of the sub-ion-scale anisotropy of the turbulence in the Earthsmagnetosheath. By measuring the five-point second-order SF s of the magnetic field, the three-dimensional structuresof the turbulence have been quantitatively characterized. Specifically, the three characteristic lengths of the eddiesare found to roughly satisfy l (cid:107) > L ⊥ > l ⊥ in the local reference frame defined by Chen et al. (2012). As for the scale-dependency of the anisotropy inferred from SF s of the total magnetic field, (1) the parallel − perpendicular anisotropyas revealed by the ratio of l (cid:107) /l ⊥ , shows an increase trend towards small scales and obeys a scaling of l (cid:107) (cid:39) . · l . ⊥ between 0.1 d i and 1 d i , then it decreases and obeys l (cid:107) (cid:39) · l . ⊥ between 0.04 d i and 0.08 d i . (2) the anisotropy inthe perpendicular plane as revealed by the ratio of L ⊥ /l ⊥ , is generally weaker than the ratio of l (cid:107) /l ⊥ . Moreover, thisanisotropy is scale-invariant, displaying a scaling of L ⊥ (cid:39) . · l . ⊥ .Interestingly, the parallel − perpendicular anisotropy tends to become increasingly isotropic when approaching bothlarge scales ∼ d i and small scales ∼ d i (Figure 4 a-c). This large-scale isotropy may reflect similar structuresas in the isotropic solar wind reported recently (i.e.,Wang et al. (2019b); Wu et al. (2019a,b)), while the small-scaleisotropy has not been reported before to our knowledge. Possible explanations for such isotropy include the weakeningof perpendicular cascade and the influence of ion cyclotron waves. Indeed, there are a few events with l ⊥ > l (cid:107) in thedatabase, where the existence of ICW has been confirmed by checking the polarisation state of the fluctuations. In afew other events, we also find coexistence of ICW and 2D l (cid:107) > l ⊥ structures through inspecting the SF s in the 2D( l (cid:107) , l ⊥ ) plane.The scaling of l (cid:107) ∝ l . ⊥ observed in this work is different from traditional KAW theory of l (cid:107) ∝ l / ⊥ , but is closeto the theoretical prediction of l (cid:107) ∝ l / ⊥ from Boldyrev & Perez (2012), Boldyrev & Loureiro (2019) and simulationfrom Cerri et al. (2019). It also corresponds to α = 1 .
13 in the framework of the model by Landi et al. (2019).Hence a modified CB premise may be needed to understand the kinetic turbulence in magnetosheath. We note that,for most of the events, the week anisotropy in the perpendicular plane is inconsistent with results of (Boldyrev &
Wang et al.
Figure 4.
Statistical analysis of the 3D anisotropy scalings. The first, second, and third column shows respectively theanisotropy of δB , δB ⊥ , and δB (cid:107) . The light grey curves represent the superposition of the statistical results, the median valueand fitted results are overplotted in bold solid and dotted lines. The dashed lines represent specifically, the reference scalingswith slope of 1/3, 2/3, 1 at top panels, and 4/2, 3/2, 1 at bottom panels. The histogram of the anisotropy scalings at 0.04 d i – 0.08 d i (light grey), and 0.1 d i – 1.0 d i (dark grey) are inserted in top panels. Similarly, the histograms of the anisotropy at0.02 d i – 1.0 d i (light grey) are inserted in bottom panels. Loureiro 2019), which predicts a much stronger anisotropy and a steeper scaling of L ⊥ ∝ l / ⊥ or L ⊥ ∝ l ⊥ , dependingon different current sheet configurations for the tearing instability. Capturing the active signatures of current sheetdisruption / reconnection (i.e., (Mallet et al. 2017; Loureiro & Boldyrev 2017)) from in-situ observation is a challengingtask, but will contribute to understand its effects on the anisotropy.The spectral anisotropy of kinetic plasma turbulence is believed to be associated with dispersion and intermittencyeffects (Zhao et al. 2016; Landi et al. 2019). For example, the anisotropic scalings are different below and above ioncyclotron frequency and also differs for sheet and tube like turbulence (Zhao et al. 2016). To illustrate the possibleconnection between intermittency and spectral anisotropy, we specifically compare the results from two events. Figure5 shows the excess Kurtosis and anisotropy relation for the total magnetic field. The excess Kurtosis is defined as K ( l ) = S ( l ) /S ( l ) −
3, where S ( l ) is the fourth-order structure function. In both panels of Figure 5, the solid linesrepresent the results from event 1, which is recorded on 4 Oct 2017 and is used as our example of spectral anisotropy insection 3, while the dash-dot lines represent the results from event 2, which is recorded on 24 Dec 2017. As plotted inthree directions l ⊥ , L ⊥ , l (cid:107) , the value of K ( l ) is around zero at large scales, meaning the roughly Gaussian distributionof the fluctuations. Toward small scales, K ( l ) displays an increase tendency before it drops down. Such non-Gaussianstatistics ( K ( l ) >
0) confirms the presence of intermittency in the magnetosheath, while the scale-dependent profileof Kurtosis is similar with solar wind observations (He et al. 2019). For the kinetic scale parallel-perpendicularanisotropy, we find that it can be considerably affected by the intermittency. As shown in Figure 5, the stronger theintermittency (see the larger Kurtosis of the solid lines in the left panel), the stronger the anisotropy level (see the largervertical distances between the solid blue curve and the grey line in the right panel). This phenomenon is consistentwith previously observations at large scales, which emphasize the key role of intermittency in generating the spectralanisotropy (i.e., (Wang et al. 2014; Pei et al. 2016; Yang et al. 2018)). For the anisotropy in the perpendicular plane,the anisotropy levels (see the two red curves in the right panel) of these two events are much smaller as compared with nisotropies and scalings k ⊥ > k (cid:107) fluctuations can be observed with non-axisymmetricfeatures in the spacecraft frame due to a sampling effect (i.e., Alexandrova et al. (2008); Turner et al. (2011); Lacombeet al. (2017) and Matteini et al. 2020, submitted). Therefore, the spectral anisotropy (non-axisymmetric) in theperpendicular plane needs be cautiously interpreted with such effects being quantitively explored in the future. Lastly,we note that both of the events still have non-Gaussian fluctuations (as seen in the non-zero values of the Kurtosis),hence the absence of isotropic l (cid:107) ≈ L ⊥ ≈ l ⊥ relation is not contradictory to previous studies, which found isotropywhen the intermittency is removed (i.e., (Pei et al. 2016)). We plan to conduct a much comprehensive analysis tounderstand how intermittency influence spectral anisotropy in a future work, particularly focusing on comparing therole of different coherent structures on the anisotropy (i.e., 2D tube-like vortices in Wang et al. (2019a) or 1D currentsheets. Figure 5.
Comparisons of the intermittency and anisotropy relation between two events during 08: 02: 13 - 08: 12: 33 on 4October 2017 and 01: 04: 43 - 01: 11: 53 on 24 December 2017.
ACKNOWLEDGMENTSWe greatly appreciate the MMS development and operations teams, as well as the instrument PIs, for data accessand support. This work was supported by the Marie Skodowska-Curie grant No. 665593 from the European UnionsHorizon 2020 research and innovation programme. J.-S. He is supported by NSFC under 41874200 and 41421003.APPENDIX A. COMPARISON OF MULTI-POINT STRUCTURE FUNCTIONSThe differences between multi-point second-order structure functions of the total magnetic field are compared here,where the two-point and three-point structure functions are defined as S (2)2 ( l ; f ) = (cid:104)| f ( r + l ) − f ( r ) | (cid:105) r , and S (3)2 ( l ; f ) =1 / (cid:104)| f ( r − l ) − f ( r ) + f ( r + l ) | (cid:105) r , respectively. As seen in the left panel of Figure 6, the trend of three-point and five-point SF tend to agree with each other, whereas the slope of two-point SF are relatively flatter, especially towardssmall scales. We also compute the “equivalent spectrum defined as S ( k ) /k and compare the results with power0 Wang et al. spectral density (PSD) from Fourier transform. Again, it is found that within [0.05, 1] km − of the right panel, theslope of the spectrum based on five-point SF is around -2.86, which is similar with the three-point result of -2.80 andthe slope of PSD around -2.94, while the slope based on two-point SF is only -2.55. Hence it is demonstrated that inorder to capture the spectral characteristics of the turbulence at sub-ion regime, the use of multi-point ( >
2) structurefunctions are preferred.
Figure 6.
Comparison of multi-point structure functions during 08:02:13-08:12:33 on 4 Oct 2017.B.
VALIDITY OF THE TAYLOR HYPOTHESISAt kinetic scales, the Taylor hypothesis may be violated by to the significant fluctuations in the turbulent flows,or due to the large phase speed of the fluctuations exceeding the bulk flow velocity (e.g. Treumann et al. (2019);Huang & Sahraoui (2019)). The validity of Taylor hypothesis for all the events is checked by comparing the structurefunction of magnetic fluctuations in two ways (Chen & Boldyrev 2017): one is to calculate the structure function fromsingle-spacecraft measurements by assuming Taylor hypothesis, and the other is to calculate the structure functionbased on direct spatial differences between measurements from six pairs of MMS spacecrafts, which are separated bycertain inter-distances between them.Figure 7 shows the statistical results of the first-order structure function as a function of scale. As representedby different colour for each individual event, the results based on Taylor hypothesis (curves) are close to the resultsfrom direct spatial measurements (crosses) at 5 km < l <
200 km. Therefore, the use of Taylor hypothesis in ouranalysis have been proven to be reasonable. We note that our results at small scales are in agreement with recentdemonstration of Taylor hypothesis being valid down to kd e ∼ k ⊥ > k (cid:107) ) in our events is also in favour of the Taylor assumptions.REFERENCES Alexandrova, O. 2008, Nonlinear Processes in Geophysics,15, 95, doi: 10.5194/npg-15-95-2008Alexandrova, O., Chen, C. H. K., Sorriso-Valvo, L.,Horbury, T. S., & Bale, S. D. 2013, SSRv, 178, 101,doi: 10.1007/s11214-013-0004-8 Alexandrova, O., Krishna Jagarlamudi, V., Rossi, C., et al.2020, arXiv e-prints, arXiv:2004.01102.https://arxiv.org/abs/2004.01102 nisotropies and scalings Figure 7.